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The Analytic Rank of a Family of Jacobians of Fermat Curves

100%

Jędrzejak T.

nr 3

199-206

We study the family of curves $F_{m}(p): x^{p} + y^{p} = m$, where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves $F_{m}(p)$. As a corollary we conclude that the jacobians of the curves $F_{m}(5)$ with even analytic rank and those with odd analytic rank are equally distributed.

A note on the torsion of the Jacobians of superelliptic curves $y^{q} = x^{p} + a$

100%

Jędrzejak T.

Banach Center Publications

2016

tom 108

nr 1

143-149

This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) $C_{q,p,a}: y^{q} = x^{p} + a$, and its Jacobians $J_{q,p,a}$, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of $J_{3,5,a}(ℚ)$ (resp. $J_{q,p,a}(ℚ)$). The main tools are computations of the zeta function of $C_{3,5,a}$ (resp. $C_{q,p,a}$) over $𝔽_{l}$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp)) and applications of the Chebotarev Density Theorem.

Congruent numbers over real number fields

100%

Jędrzejak T.

nr 2

179-186

It is classical that a natural number n is congruent iff the rank of ℚ -points on Eₙ: y² = x³-n²x is positive. In this paper, following Tada (2001), we consider generalised congruent numbers. We extend the above classical criterion to several infinite families of real number fields.

Characterization of the torsion of the Jacobian of y² = x⁵+Ax and some applications

63%

Jędrzejak T. , Ulas M.

Acta Arithmetica

2010

tom 144

nr 2

183-191

Cubic forms, powers of primes and the Kraus method

51%

Dąbrowski A. , Jędrzejak T. , Krawciów K.

nr 1

35-48

We consider the Diophantine equation $(x+y)(x²+Bxy+y²) = Dz^{p}$, where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D's, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).

Tuples of hyperelliptic curves y² = xⁿ+a

51%

Jędrzejak T. , Top J. , Ulas M.

tom 150

nr 2

105-113